Question

Given the parent function y=x2
write the equations with the following transformations in the form y=a(x−h)2+k
.

A. The graph has been shifted to the left 1 and up 5

B. The graph has been reflected over the x-axis and stretched vertically by a factor of 3.

C. The graph has been compressed vertically by a factor of 1/2 and shifted right 6

(6 points)

Answer 1

To write the equations with the specified transformations of the parent function \( y = x^2 \) in the form \( y = a(x - h)^2 + k \), we can analyze each transformation:

### A. Shifted left 1 and up 5
When we shift the graph to the left by 1, we replace \( x \) with \( x + 1 \). Shifting it up by 5 means adding 5 to the function.

Starting with the parent function:
\[
y = x^2
\]

Transforming:
\[
y = (x + 1)^2 + 5
\]

The final equation is:
\[
\boxed{y = (x + 1)^2 + 5}
\]

### B. Reflected over the x-axis and stretched vertically by a factor of 3
Reflecting over the x-axis means we multiply the function by -1. Stretching vertically by a factor of 3 means multiplying the entire function by 3.

Starting with the parent function:
\[
y = x^2
\]

Transforming:
\[
y = -3(x^2)
\]
(Note: We can write this as \(y = -3(x - 0)^2 + 0\) for the specific form requested.)

The final equation is:
\[
\boxed{y = -3(x - 0)^2 + 0}
\]

### C. Compressed vertically by a factor of \( \frac{1}{2} \) and shifted right 6
When compressing the graph vertically by a factor of \( \frac{1}{2} \), we multiply the function by \( \frac{1}{2} \). Shifting it right by 6 means replacing \( x \) with \( x - 6 \).

Starting with the parent function:
\[
y = x^2
\]

Transforming:
\[
y = \frac{1}{2}((x - 6)^2)
\]

The final equation is:
\[
\boxed{y = \frac{1}{2}(x - 6)^2}
\]

In summary, the equations are:
A. \( y = (x + 1)^2 + 5 \)
B. \( y = -3(x - 0)^2 + 0 \)
C. \( y = \frac{1}{2}(x - 6)^2 \)